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| Mirrors > Home > MPE Home > Th. List > prnz | Structured version Visualization version GIF version | ||
| Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.) |
| Ref | Expression |
|---|---|
| prnz.1 | ⊢ 𝐴 ∈ V |
| Ref | Expression |
|---|---|
| prnz | ⊢ {𝐴, 𝐵} ≠ ∅ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | prnz.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 2 | 1 | prid1 4762 | . 2 ⊢ 𝐴 ∈ {𝐴, 𝐵} |
| 3 | 2 | ne0ii 4344 | 1 ⊢ {𝐴, 𝐵} ≠ ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 {cpr 4628 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: opnz 5478 propssopi 5513 fiint 9366 fiintOLD 9367 wilthlem2 27112 upgrbi 29110 wlkvtxiedg 29643 shincli 31381 chincli 31479 constrextdg2lem 33789 spr0nelg 47463 sprvalpwn0 47470 |
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